\newproblem{lay:6_1_26}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 6.1.26}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $\mathbf{u}=\begin{pmatrix}5\\-6\\7\end{pmatrix}$, and let $W$ the set of all $\mathbf{x}\in\mathbb{R}^3$ such that $\mathbf{u}\cdot\mathbf{x}=0$.
	What theorem of Chapter 4 can be used to show that $W$ is a subspace of $\mathbb{R}^3$? Describe $W$ in geometric language.
}{
   % Solution
	We may use Theorem 4.2.2 in which it is stated that the null space of an $m\times n$ matrix is a subspace of $\mathbb{R}^n$. We simply need to use the matrix
	$A=\mathbf{u}^T$. Its null space is formed by all those vectors such that
	\begin{center}
		$A\mathbf{x}=\mathbf{u}^T\mathbf{x}=\mathbf{u}\cdot\mathbf{x}=0$
	\end{center}
	Geometrically, $W$ is formed by the plane through the origin and perpendicular to the vector $\mathbf{u}=\begin{pmatrix}5\\-6\\7\end{pmatrix}$.
}
\useproblem{lay:6_1_26}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
